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Uncertainty propagation in nuclear forensics
Applied Radiation and Isotopes 89 (2014) 58–64
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Uncertainty propagation in nuclear forensics S. Pommé a,n, S.M. Jerome b, C. Venchiarutti a a b
European Commission, Joint Research Centre, Institute for Reference Materials and Measurements, Retieseweg 111, B-2440 Geel, Belgium National Physical Laboratory, Teddington, Middlesex TW11 OLW, UK
H I G H L I G H T S
Uncertainty propagation formulae for age dating with nuclear chronometers. Applied to parent–daughter pairs used in nuclear forensics. Investigated need for better half-life data.
art ic l e i nf o
a b s t r a c t
Article history: Received 29 November 2013 Received in revised form 31 January 2014 Accepted 6 February 2014 Available online 14 February 2014
Uncertainty propagation formulae are presented for age dating in support of nuclear forensics. The age of radioactive material in this context refers to the time elapsed since a particular radionuclide was chemically separated from its decay product(s). The decay of the parent radionuclide and ingrowth of the daughter nuclide are governed by statistical decay laws. Mathematical equations allow calculation of the age of specific nuclear material through the atom ratio between parent and daughter nuclides, or through the activity ratio provided that the daughter nuclide is also unstable. The derivation of the uncertainty formulae of the age may present some difficulty to the user community and so the exact solutions, some approximations, a graphical representation and their interpretation are presented in this work. Typical nuclides of interest are actinides in the context of non-proliferation commitments. The uncertainty analysis is applied to a set of important parent–daughter pairs and the need for more precise half-life data is examined. & 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).
Keywords: Nuclear forensics Uncertainty Half-life Decay data Accuracy Non-proliferation
1. Introduction Age dating of radioactive material by means of radiometric or mass spectrometry measurements is a long-established technique in geological and archaeological sciences (Hamilton, 1965; Magill and Galy, 2005). It uses statistical decay laws to provide the link between the activity or atom concentration of radionuclides and their daughter nuclide(s) and the time elapsed since certain initial conditions. Nuclear forensics is a relatively young scientific discipline in which age dating is performed using the same principles. Here, the age refers to the time elapsed since a radionuclide of interest, usually an actinide isotope, was chemically separated from its decay products (Mayer et al., 2007; 2013). Such information taken with other evidence is crucial for identification of the sampled material (see e.g. Schwantes et al. (2009)). The precision by which the age of the material can be determined not only depends on the precision of the analytical measurement,
n
Corresponding author. Tel.: þ 32 14 571 289; fax: þ 32 14 571 864. E-mail address: [email protected] (S. Pommé).
but is also limited by the state of knowledge of the half-lives involved. In this paper, exact mathematical formulae are presented of the uncertainty propagation factors involved with age dating in nuclear forensics, based on the measurement of either atom ratios or activity ratios. From the exact equations, approximate formulae are derived which are applicable under specified conditions. Graphs of the uncertainty propagation as a function of time for two hypothetical cases are presented and interpreted. The equations are used specifically to examine the constraints imposed on the attainable precision of the age due the current uncertainties on the nuclear half-lives of the parent–daughter pairs involved.
2. Age dating by atom ratio measurements 2.1. Exact formulae Age dating is applied to a material containing a radionuclide and its decay products, assuming that no physicochemical processes other than radioactive decay have altered their relative
http://dx.doi.org/10.1016/j.apradiso.2014.02.005 0969-8043 & 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).
S. Pommé et al. / Applied Radiation and Isotopes 89 (2014) 58–64
concentrations over time. The number of atoms of the parent and daughter nuclides, P(t) and D(t) respectively, obey statistical decay rules as a function of time: PðtÞ ¼ Pð0Þe λP t DðtÞ ¼ Dð0Þe λD t þ Pð0Þ
ð1Þ λP ðe λP t e λD t Þ λD λP
ð2Þ
in which the decay constants λP and λD are inversely proportional to the parent and daughter half-lives, TP ¼ln(2)/λP and TD ¼ln(2)/λD, respectively. The ratio of Eq. (2) and Eq. (1) gives the equation for the atom ratio of daughter and parent atoms at any time t40: RðtÞ ¼
DðtÞ λP ¼ Rð0Þe ðλD λP Þt þ ð1 e ðλD λP Þt Þ PðtÞ λD λP
ð3Þ
It is assumed that at time t¼0, the parent nuclei were separated completely from the daughter nuclei such that R(0)¼0, so that D(t) can be fully ascribed to ingrowth after t¼0. Determining the ‘age’ of the material consists of calculating the most likely amount of time elapsed since separation, using measured values of the atom ratio, ^ RðtÞ, and literature values of the decay constants, λ^ P and λ^ D : ! 1 λ^ D λ^ P ^ Age ¼ ln 1 RðtÞ ð4Þ λ^ P λ^ D λ^ P Linear propagation of uncertainty on the atom ratio R(t) and the decay constants λP and λD results to: 2 sðtÞ 2 λD T λP sðλP Þ 2 ¼ t λP λP λD t λD 2 λD T sðλD Þ 2 1 þ λD λP λD t 2 2 T sðRÞ þ ð5Þ t R in which the variable T is defined as T eðλD λP Þt 1 ¼ t ðλD λP Þt
ð6Þ
and the relative uncertainties on the decay constants are equal to (minus) the relative uncertainties on the half-lives, i.e. sðλÞ=λ ¼ sðT 1=2 Þ=T 1=2 : The uncertainty on R arises mainly from the uncertainty of the analytical measurement result, which may be expected to be smallest around R¼1 and to increase in approximate proportion to |log(R)| when parent and daughter concentrations differ by orders of magnitude. Another uncertainty component that should be taken into account is the possibility of an incomplete chemical separation at time t¼ 0 (Williams and Gaffney, 2011; Eppich et al., 2013). This would increase R(t) by a relative amount Rð0Þ=RðtÞe ðλD λP Þt a0 (Eq. (3)), which also propagates by a factor T/t to the relative uncertainty of the age estimate (Eq. (5)).
the daughter nuclide, R(TD) EλP/2λD. As a result, the expectation value of the daughter concentration remains smaller than that of the parent concentration. In the case of a long-lived daughter, where λD⪡λP, the atom ratio reaches unity after about one half-life of the parent, R(TP)E1, and then increases exponentially by RðtÞ eλP t for λPt 41. 2.2.2. Uncertainty propagation The uncertainty propagation formulae in Eq. (5) can be stated as a series expansion, resulting in approximate formulae valid for |λD λP|to1: 2 sðtÞ 2 λD t λP ðλD λP Þt 2 sðλP Þ 2 1 t 2 λP 6 2 2 2 λD t λP ðλD λP Þt sðλD Þ þ þ 2 λD 6 !2 2 2 ðλD λP Þt ðλD λP Þ t sðRÞ 2 þ þ 1þ ð7Þ 2 R 6
In Table 1, approximate equations are summarised for the propagation factors under boundary conditions. For material with a relatively young age compared to the half-lives involved, λPt⪡1 and λDt⪡1, the propagation factors for λP and R are unity, while the factor for λD is insignificantly small. Consequently, age determinations based on atom ratios are more sensitive to the parent half-life then to the daughter half-life. This is compatible with the fact that R(t) EλPt for small values of t, i.e. linear with λP and independent of λD. Only for old material, |λD λP|t b 1, does the uncertainty propagation of the daughter half-life become important, but under these conditions the over-all accuracy of the method is relatively poor, as the propagation factors increase almost exponentially with λDt. 2.3. Graphical representation The propagation factors are presented for two hypothetical cases: a long-lived parent nuclide (λP⪡λD) in Fig. 1 and a long-lived daughter nuclide (λD⪡λP) in Fig. 2. 2.3.1. Long-lived parent In the top graph of Fig. 1, there is no visible difference between the dominant propagation factors of λP and R. They remain close to unity as long as λDto1. For larger values of t, the propagation factors increase exponentially with λDt. The age dating method Table 1 Exact and approximate equations for uncertainty propagation factors for age dating via atom ratio measurements in specific conditions. The variable T has been defined in Eq. (6). Exact equations are also presented as a function of the half-lives TP and TD instead of the decay factors λP and λD. Condition
2.2. Approximate equations Always valid
2.2.1. Atom ratio Serial expansion of Eq. (3) shows that R(t) EλPt if the age of the material is low compared to the half-lives of the nuclides involved (|λD λP|t o1). This means that the initial ingrowth of the daughter happens at the rate of the decay of the parent and is, to a first order approximation, independent of the half-life of the daughter. Further changes to R(t) with time depend on which nuclide has the longest half-life. In the case of a long-lived parent, where λP⪡λD, the atom ratio tends towards the secular equilibrium value R(t)E λP/λD for λDt 41 and reaches about half that value after one half-life of
59
Always valid
Age from atom ratio, using Eq. (3) Propagation factor ∂t λP ∂λP t λD T λP λP λD t λD TP T D T P Tt TT DP
λP { λD λD t{1
Long-lived parent 1
λD t c 1 λP c λD λP t{1
eλD t λD t Long-lived daughter 1
λP t c 1
1
∂t λD ∂λD t λD T λP λD t 1 T P T T D T P t 1
∂t R
λD t 2 eλD t λD t
1
λD t 2 λD λP
∂R t
T t
T t
eλD t λD t 1 1 λP t
60
S. Pommé et al. / Applied Radiation and Isotopes 89 (2014) 58–64
Fig. 1. Hypothetical case of a long-lived parent nuclide with 100 times longer halflife than its daughter nuclide. Top: Propagation factors that relate relative uncertainty of age dating by means of atom ratio measurements with relative uncertainties on the analytical result and on the decay constants of the parent and daughter nuclide (Eq. (5)). Middle: Uncertainty propagation factors for activity ratio measurements (Eq. (12)). Bottom: Order of magnitude of the atom and activity ratio (full line) and indicator of uncertainty on the analytical result (cf. Eq. (15)) multiplied with the propagation factor T/t (dotted line). The latter can be used to roughly identify a time region in which age dating can be performed accurately.
breaks down because the atom ratio between parent and daughter concentration tends towards the secular equilibrium value of λP/(λD λP). Under these conditions, where λDt⪢1, the method can only establish a minimum age value, i.e. Age⪢1/λD. 2.3.2. Long-lived daughter The top graph of Fig. 2 refers to the case of a long-lived daughter nuclide (λD⪡λP). The propagation factor of λP is about 1 and that of λD is significantly lower, with values varying between λDt/2 (for λPto1) and λD/λP (for λPt41). The propagation factor for R is close to 1 for λPto1 and decreases quasi-linearly towards 1/λPt for λPt41. The method seemingly gains accuracy with time; however, one also has to take the uncertainty s(R) of the atom ratio into account. After about one half-life of the parent nuclide, the concentration of the daughter nuclide becomes greater than that of the parent nuclide and one expects the uncertainty of the analytical measurement, s(R)/R, to increase as this ratio grows exponentially with λPt over many orders of magnitude. The method will break down at a point where the ratio R becomes too large to be measurable. The precision of the age may at all times be limited by the level of uncertainty on the half-life of the parent.
Fig. 2. Hypothetical case of a short-lived parent nuclide with 100 times shorter half-life than its daughter nuclide. Top: Propagation factors that relate relative uncertainty of age dating by means of atom ratio measurements with relative uncertainties on the analytical result and on the decay constants of the parent and daughter nuclide (Eq. (5)). Middle: Uncertainty propagation factors for activity ratio measurements (Eq. (12)). Bottom: Order of magnitude of the atom and activity ratio (full line) and indicator of uncertainty on the analytical result (cf. Eq. (15)) multiplied with the propagation factor T/t (dotted line). The latter can be used to roughly identify a time region in which age dating can be performed accurately.
constants, i.e. AP(t)¼P(t)λP and AD(t)¼D(t)λD. The time dependence follows from the statistical decay rules: AP ðtÞ ¼ AP ð0Þe λP t
ð8Þ
λD AD ðtÞ ¼ AD ð0Þe λD t þ AP ð0Þ ðe λP t e λD t Þ λD λP
ð9Þ
and the activity ratio of a parent–daughter pair at time t is RA ðtÞ ¼
AD ðtÞ λD ¼ RA ð0Þe ðλD λP Þt þ ð1 e ðλD λP Þt Þ AP ðtÞ λD λP
ð10Þ
Again one assumes full separation between parent and daughter nuclides at time t ¼0, such that RA(0) ¼0. The ‘age’ of the material can be calculated from a measured activity ratio value, R^ A ðtÞ, and best estimates of the decay constants, λ^ P and λ^ D : ! 1 λ^ D λ^ P Age ¼ ln 1 R^ A ðtÞ ð11Þ λ^ P λ^ D λ^ D
3. Age dating by activity ratio measurements 3.1. Exact equations The activities of the parent and daughter nuclides in a material are proportional to their respective number of atoms and decay
The uncertainty propagation of the concentration ratio RA(t) and the decay constants λP and λD to the age is given by 2 sðtÞ 2 λP T sðλP Þ 2 1 ¼ t λP λP λD t
S. Pommé et al. / Applied Radiation and Isotopes 89 (2014) 58–64
λP T λD sðλD Þ λD λP λD t λP T sðRA Þ þ t RA
3.3. Graphical representation
þ
ð12Þ
in which the variable T is the same as in Eq. (6). The uncertainty on RA mainly arises from the uncertainty of the activity ratio measurement result, which may be expected to be at its smallest around RA ¼ 1 and to grow roughly proportional with |log(RA)| when parent and daughter activities differ by orders of magnitude. 3.2. Approximate equations 3.2.1. Activity ratio For small values of t, where |λD λP|to1, the activity ratio in Eq. (10) reduces to RA(t) EλDt. This is a consequence of the fact that the atom ratio grows linearly with time (R(t)EλPt) and the specific activity ratio between same amounts of daughter and parent nuclide is λD/λP. The rate at which the daughter activity initially grows with time is, to a first order approximation, independent of the half-life of the parent. Old material, in the case of a long-lived parent (λP⪡λD), tends towards secular equilibrium in which parent and daughter activities are nearly the same, RA E1, whereas in the case of a long-lived daughter (λD⪡λP), the activity ratio increases exponentially, as RA ðtÞ λD =λP eλP t for λPt41. 3.2.2. Uncertainty propagation The uncertainty propagation formulae in Eq. (12) for age dating by activity ratio can also be stated as a series expansion: 2 sðtÞ 2 λP t λP ðλD λP Þt 2 sðλP Þ 2 t 2 λP 6 2 λP t λP ðλD λP Þt 2 sðλD Þ 2 þ þ 1 þ 2 λD 6 !2 2 2 ðλD λP Þt ðλD λP Þ t sðRA Þ 2 þ þ 1þ ð13Þ 2 RA 6 In Table 2, approximate equations are summarised for the propagation factors under boundary conditions. For material with a young age compared to the half-lives involved, λPt⪡1 and λDt⪡1, the propagation factors for λD and RA are unity, while the factor for λP is insignificantly small. Consequently, age determinations based on activity ratios are more sensitive to the daughter half-life then to the parent half-life, which is the opposite of the effect noticed for atom ratios (Eq. (7)). Table 2 Exact and approximate equations for uncertainty propagation factors for age dating via activity ratio measurements in specific conditions. The variable T has been defined in Eq. (6). Exact equations are also presented as a function of the half-lives TP and TD. Condition
Always valid Always valid λP {λD λD t{1 λD t c 1 λP cλD λP t{1 λP t c 1
Age from activity ratio, using Eq. (11) Propagation factor ∂t λP ∂λP t λP T λP λD t 1 T D T T D T P t 1 Long-lived parent λP t 2 λP eλD t λD λD t Long-lived daughter
61
∂t λD ∂λD t λP T λD λP λD t λP TD T TP T D T P t T D
∂t RA ∂RA t T
1
1 λD t
t
T t
λP e λD λD t
eλD t λD t
λP t 2
1
1
1
1 λP t
1 λP t
To interpret the impact of various parameters on the over-all uncertainty of age dating via the activity ratio, one can refer again to Figs. 1 and 2 representing two hypothetical cases: a long-lived parent nuclide, when λP⪡λD, in Fig. 1 and a long-lived daughter nuclide, when λD⪡λP, in Fig. 2. 3.3.1. Long-lived parent In the middle graph of Fig. 1, the dominant propagation factors of λD and RA remain close to unity so long as λDto 1. The propagation factor of λP is much smaller, but increases exponentially for λDt⪢1 and eventually its uncertainty propagation becomes as important as that of λD, but inferior by a factor of λP/ λD to the propagation factor of RA. For large values of t, the method can only establish a minimum age value, i.e. Age⪢1/λD. 3.3.2. Long-lived daughter The middle graph of Fig. 2 refers to the case of a long-lived daughter nuclide, (λD⪡λP). The propagation factors of RA and λD are nearly equal: close to unity for λPto 1 and linearly decreasing with 1/λPt for λPt 41. The propagation factor of λP is small for λPto 1 and tends to unity for λPt 41.
4. Atom versus activity ratio 4.1. Indicator of accuracy It is interesting to note that the activity ratio can differ significantly from the atom ratio if parent and daughter nuclides have distinctly different half-lives. The choice of the most suitable measurement technique for a given parent–daughter pair may be evaluated by comparison through the relationship log ðRA Þ ¼ log λD þ log ðRÞ ð14Þ λP The activity ratio measurements inflate the signal of the shortlived nuclide, which is an analytical advantage in the case of a long-lived parent (log(λD/λP) 40), to compensate for the relatively low concentration of a short-lived daughter (log(R) o0). It offers also potential with aged material with a short-lived parent (log(λD/ λP) 40), to amplify the signal of its remaining small traces (log(R) 40). The bottom graphs in Figs. 1 and 2 show the logarithmic value of R and RA, as well as a new quantity:
1 T T log 2þRþ j log ðRÞj þ log ð15Þ R t t which is assumed to be roughly indicative of the relative uncertainty s(R)/R and its uncertainty propagation factor T/t towards the age. It was derived from an assumption that the signals from D and P are Poissonian, such that the relative uncertainty on R would be inversely proportional with the count rates, i.e. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi log ðsðRÞ=RÞ p log ð 1=P þ 1=DÞ p log ðð1 þ RÞ þ ð1 þ 1=RÞÞ, since P/(P þD) ¼1/(1 þ R) and D/(Pþ D)¼R/(1þR). A similar quantity for the activity ratio RA was plotted in Figs. 1 and 2. The lower values of these quantities roughly indicate the time range in which the age dating methods should be relatively accurate. 4.2. Long-lived parent In Fig. 1, the activity ratio is closer to unity than the atom ratio over the whole time range, which in principle is advantageous. Whether activity measurements can lead to a more accurate age
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S. Pommé et al. / Applied Radiation and Isotopes 89 (2014) 58–64
determination will depend on the difference in attainable precision by radiometric methods and mass spectrometry, as well as the difference in accuracy of the half-lives of parent and daughter nuclide involved. The long half-life of the parent implies a low specific activity, which may be problematic when measuring the activity of small amounts of material. In the bottom graph of Fig. 1, the time region in which both age dating methods are likely to be more accurate roughly corresponds to ages varying within 1/10 and 3 times the half-life of the daughter nuclide TD. 4.3. Long-lived daughter In the bottom graph of Fig. 2, the most favourable time regions of both methods differ significantly. Atom ratio measurements should be informative over a wide time region, for example between 10 3 and 10 times the half-life of the parent nuclide TP. Activity ratios are less favourable chronometers for this type of decay, unless for old material, where the age is between 1/3 and 12 times TP.
half-lives are often unrealistically low (Pommé, 2007; Pommé et al., 2008). As an alternative to atom ratio measurements, activity measurements offer – at least in principle – a lower minimum uncertainty for certain chronometers, like 232U–228Th, 238 U–234Th, 237Np–233U, 242Pu–238U and 243Am–239Pu. Whether this can also be realised in practice heavily depends on the achievable precision on the activity ratio, as the radiometric method may have a lower dynamic range than mass spectrometry. The reader is referred to Mayer et al. (2013) for a discussion of measurement techniques for U and Pu chronometers. 5.3. Other decay types 5.3.1. Stable daughter In Table 3, one finds decay types that do not strictly match the parent–daughter decay model investigated in this work. Some of the daughter nuclides are stable (λD ¼0) and the equation for the age dating via atom ratio reduces to: Age ¼
1 ^ lnð1 þ RðtÞÞ λ^ P
ð16Þ
5. Specific radionuclide chronometers 5.1. Nuclear chronometers Nuclear forensics refers to the scientific analysis of radioactive material in the context of legal procedures (Mayer et al., 2013), and is no different in concept than any other forensic investigation. One may think of material used in the construction of improvised nuclear devices (IND) or radiation dispersion devices (RDD) that may be detonated in populated areas. Age dating is part of characterisation of nuclear material, which may play a role in establishing the origin of intercepted material, identifying perpetrators and their network, as well as providing evidence to bring them to justice. Given the severity of such activities and the possibly long-reaching consequences, it is imperative that all parts of the forensic analysis comply with the accepted norms for the presentation of scientific evidence in court (United States Appeal Court, 1995). The chronometers of most interest are actinides arising from their potential use in INDs, which includes isotopes of thorium (Z¼ 90), uranium (Z¼92), neptunium (Z¼93), plutonium (Z¼94) and americium (Z ¼95), whereas candidate materials for RDDs may include radionuclides such as 60Co, 89Sr, 90Sr and 137Cs. Table 3 shows a non-exhaustive list of nuclear chronometers, the evaluated half-lives of the parent–daughter pairs (DDEP, 2004– 2013), the uncertainty propagation factors (cf. equations in Tables 1 and 2) for material of age 1 year and 50 years, and a minimum uncertainty value based on the propagation of the halflife uncertainties only. 5.2. Objective If one sets as an objective to keep the uncertainty on age dating below 50 days over a range of 1–50 years, which corresponds to a relative uncertainty of 0.27% (k ¼1), then the relative uncertainties on the ratio measurements and the half-lives should be below 0.25%. In Table 3, the dominant uncertainty component is indicated in bold when assuming hypothetically that the uncertainty on the ratio measurement is 0.25%. Any nuclide of which the propagation factor is in bold therefore would need a more accurate half-life value to achieve the objective of s(Age) o50 days. This includes a majority of the nuclides in the list, of which 232 U, 229Th and 242mAm are obvious examples. Moreover, the list may in reality be longer, as the uncertainty estimates on measured
Obviously the relative uncertainty on the half-life of the parent is propagated linearly to the age, hence the propagation factor for λP is unity for 60Co, 89,90Sr and 137Cs (cf. Table 3). 5.3.2. Parent–daughter–granddaughter In some other cases, the decay type is more complex and involves three subsequent decays. Examples of parent–daughter– granddaughter decays are 90Sr–90Y–90Zr, 237Np–233Pa–233U, 242m Am–242gAm–242Pu and 243Am–239Np–239Pu. In these specific cases, the half-life of the daughter – 2.7 d (90Y), 27 d (233Pa), 16 h (242gAm) and 2.4 d (239Np) – is very small compared to that of the parent and granddaughter. As a result, they hardly delay the decay from parent to granddaughter and the parent–granddaughter pair can be regarded as a simple parent–daughter decay when λDt⪢1. This can be verified from the statistical equation for the atom concentration of the granddaughter nuclide: λD GðtÞ ¼ Gð0Þe λG t þ Dð0Þ ðe λD t e λG t Þ λG λD λP λD λP λD λP λD e λP t þ e λD t þ e λG t þ Pð0Þ λD λP λG λP λP λD λG λD λP λG λD λG ðGð0Þ þ Dð0ÞÞe λG t þPð0Þ
λP λD t λP e þ Pð0Þ ðe λP t e λG t Þ λD λG λP
ð17Þ
The concentration D(t) of the short-lived daughter is always small compared to P(t); their atom ratio is proportional to λP/λD. The concentration of the granddaughter, G(t), is hardly influenced by the daughter, as the term with λD is proportional to λP/λD and contains an exponential factor which is small for λDt41. Thus, for ages larger than the (short) half-life of the daughter, Eq. (17) reduces to Eq. (2) in which the granddaughter takes the place of the daughter. The uncertainty propagation of λD to the determined age is negligible. 5.4. Recommended data 5.4.1. Data evaluation Since the radionuclide community has to consider, use and thus recommend the half-lives of many different radionuclides, considerable effort has gone into evaluating nuclear decay data for an increasing number of radionuclides under the Decay Data Evaluation Project (DDEP, 2004-2013). Where a radionuclide is not yet included in the DDEP database, the usual recommendation is to
S. Pommé et al. / Applied Radiation and Isotopes 89 (2014) 58–64
63
Table 3 Parent–daughter pairs usable as nuclear chronometers, their half-lives and propagation factors for the relative uncertainty on age dating in a range of 1–50 years. Most halflife data are taken from DDEP (2004-2013) and some from ENSDF (2013). Propagation factors in bold indicate a dominant uncertainty component in the hypothetical case that s(R)/R ¼ s(RA)/RA ¼0.25%. The minimum uncertainties on the age dating, umin(days), after 1 year and after 50 years are calculated taking into account only the uncertainties on the half-lives (adopting s(R)/R¼ 0%.). Parent A
Z
60
Co Sr 90 Sr 137 Cs 230 Th 232 Th 232 U 233 U 234 U 235 U 236 U 238 U 237 Np 236 Pu 238 Pu 239 Pu 240 Pu 241 Pu 242 Pu 241 Am 242m Am 243 Am 89
n
Daughter T 1=2 ðaÞ
sðT 1=2 Þ
5.2711 (8) 0.13846 (8) 28.80 (7) 30.05 (8) 7.538 (30) 104n 1.402 (6) 1010 70.6 (1.1) 1.592 (2) 105n 2.455 (6) 105 7.04 (1) 108 2.343 (6) 107n 4.468 (5) 109 2.144 (7) 106 2.858 (8)n 87.74 (3) 24100 (11) 6561 (7) 14.33 (4) 3.73 (3) 105 432.6 (6) 143 (2) 7367 (23)
0.015 0.06 0.24 0.27 0.4n 0.43 1.6 0.13n 0.24 0.14 0.26n 0.11 0.33 0.28n 0.034 0.05 0.11 0.28 0.8 0.14 1.4 0.31
T 1=2
(%)
A
Z
60
Ni Y 90 Zr 137 Ba 226 Ra 228 Ra 228 Th 229 Th 230 Th 231 Pa 232 Th 234 Th 233 U 232 Th 234 U 235 U 236 U 241 Am 238 U 237 Np 242 Pu 239 Pu 89
Age via atom ratio R T 1=2 ðaÞ
sðT 1=2 Þ
1 1 1 1 1600 (7) 5.75 (4) 1.9126 (9) 7.34 (16) 103n 7.538 (30) 104n 3.267 (26) 104 1.402 (6) 1010 6.600 (8) 10 2 1.592 (2) 105n 1.402 (6) 1010 2.455 (6) 105 7.04 (1) 108 2.343 (6) 107 423.6 (6) 4.468 (5) 109 2.144 (7) 106 3.73 (3) 105 24100 (11)
– – – – 0.44 0.70 0.05 2.2n 0.4n 0.8 0.43 0.12 0.13n 0.43 0.24 0.14 0.26 0.14 0.11 0.33 0.8 0.8
T 1=2
(%)
Age via activity ratio RA
Combined
λP
λD
umin(d)
λP
λD
umin(d)
R, RA
1 1 1 1 1 1–70 1.2–1 1 1 1 1 3500–1 1 1 1 1 1 1 1 1 1 1
– – – – 0–0.01 0.06–70 0.2–1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.1–3 0.2–11 0.9–44 1–49 1.5–74 2–1E4 7–1 0.5–23 0.9–45 0.5–26 0.9–47 2E3–1 1–60 1–51 0.1–6 0.2–8 0.4–20 1–52 3–147 0.5–25 5–255 1–57
– – – – 0 0 0–1 0 0 0 0 0 0 0.1–0.9 0–0.2 0 0 0–0.6 0–150 0 0–0.1 0
– – – – 1 1–70 1–1 1 1 1 1 1 1 0.9–0.08 1–0.8 1 1 1–0.4 1 1 1–0.9 1
– – – – 2–80 3–127 0.2–1 8–398 1.5–73 3–145 2–78 0.5–1 0.5–23 1–47 0.9–37 0.5–26 0.9–47 0.5–34 0.4–20 1–57 3–133 0.2–8
0.9–0.15 0.2–0.004 1–0.6 1–0.6 1 1–70 1.2–1 1 1 1 1 3500–1 1 0.9–0.08 1–0.8 1 1 1–0.4 1 1 1–0.9 1
Data from ENSDF (2013).
seek data from the Evaluated Nuclear Structure File database, also published on line (ENSDF, 2013). The aim of evaluation of data is to provide the best estimate of the value of the half-live and a realistic estimate of its uncertainty.
Table 4 Comparison of half-life values and standard uncertainties (k ¼ 1) in geo- and cosmochronology and in radionuclide metrology. In the case of 230Th and 234U, recent data were included for a discussion of uncertainties (see Section 5.4.4). Nuclide
5.4.2. User community There is, however, some difference between the half-life data sets used by the radionuclide metrology community and the geochemistry community. By comparison, the geochemistry community have a smaller list of radionuclides to consider and has an interest in using a fixed, common data set for reasons of traceability and comparability of age dating results among various studies, with constraints on consistency and ‘…all selected values are open to and should be the subjects of continuing critical scrutinizing and laboratory investigation…’ (Steiger and Jäger, 1977). A partial comparison between the two sets of recommended data is given in Table 4, in which we included data from Steiger and Jäger (1977) and recently measured half-life values for 234U and 230Th (Cheng et al., 2013). The half-life values are fairly consistent, as they often rely on the same key measurements. This is for instance the case with 235,238U, which predominantly relies on the work of Jaffey et al. (1971). However, there is an essential difference in the treatment of uncertainties. Data evaluators have a more critical approach to the low uncertainties claimed in the past, resulting in a much higher uncertainty estimate (Schön et al., 2004; DDEP, 2004-2013). These evaluated data have not been applied in the geochronology community and therefore the low uncertainty estimates of Jaffey et al. (1971) have been propagated to daughter nuclides 234U and 230Th. 5.4.3. Problems with uncertainty Considering that nuclear forensics will be subject to scrutiny in a court of law, it is essential that it relies on the best available data set and in particular that it applies realistic uncertainties to avoid overly optimistic claims. One needs to be aware of problems
Geo- and cosmochronology T 1=2 ðaÞ
40
K Th 232 Th 234 U 235 U 238 U 230
1.2505 109a 7.5584 (55)103c 1.40100 1010a 2.45620 (13) 105c 7.03810 108a 4.46831 109a
sðT 1=2 Þ T 1=2
0.07 0.05
(%)
Radionuclide metrology T 1=2 ðaÞ
sðT 1=2 Þ
1.2504 (30) 109b 7.538 (30) 104d 1.402 (6) 1010a 2.455 (6) 105b 7.04 (1) 108b 4.468 (5) 109b
0.24 0.40 0.43 0.24 0.14 0.11
T 1=2
(%)
a
Steiger and Jäger (1977)—no stated uncertainty. DDEP (2004-2013). Cheng et al. (2013). d ENSDF (2013). b c
related to the assignment of uncertainties to half-lives (Pommé, 2007; Croft et al., 2013) and it is essential that future publications of new decay data would include a realistic uncertainty calculation and apply a complete reporting style to improve consistency of data sets and achieve traceability on how they were produced (Pommé et al., 2008). One of the minimum requirements is to provide a table with the uncertainty budget, i.e. a list of all possible uncertainty components, their estimated value and propagation to the half-life uncertainty. The accompanying text should clarify how these uncertainties were calculated or estimated.
5.4.4. Example Cheng et al. (2000; 2013) determined the half-lives of 234U and 230 Th twice, by measuring atom ratios in old materials and assuming that both daughter nuclides are in secular equilibrium with their long-lived parent 238U. They calculated relative uncertainty values of 0.053% (k¼ 1) and 0.073% (k ¼1) for T1/2(234U) and
64
S. Pommé et al. / Applied Radiation and Isotopes 89 (2014) 58–64
T1/2(230Th), respectively, which are dominated by the 0.054% relative uncertainty estimate on T1/2(238U) by Jaffey et al. (1971). The uncertainty estimates on the 234U/238U and 230Th/238U atom ratios are lower: 0.017% (k¼ 1) and 0.053% (k ¼1), respectively. However, their atom ratio results from 2013 differ by 0.14% and 0.14% with the ratios that they published in 2000, with uncertainties of 0.086% (k¼ 1) and 0.050% (k ¼1). The problem arises that the stated uncertainties of the 234U and 230Th may be underestimated, since: (1) the uncertainty on T1/2(238U) may be unrealistic (Schön et al., 2004; DDEP, 20042013), (2) the uncertainty propagation should have yielded relative uncertainties of 0.056% and 0.076% instead of 0.053% and 0.073%, (3) the uncertainties on the atom ratio values are barely sufficient to cover differences of 0.14% between data obtained in 2000 and 2013 by the same authors, (4) there may be additional systematic errors involved that do not show up by repetition of the same type of measurement. 6. Conclusions Exact and approximate uncertainty propagation factors have been derived mathematically for age dating of nuclear material via atom and activity ratio measurements. They are summarised in Tables 1 and 2 and graphically presented in Figs. 1 and 2. If the age of the material is low compared to the half-life of the nuclides involved, the mathematical equations simplify: (1) the relative uncertainty of the atom or activity ratio measurements propagate by a factor of 1, (2) and so does the relative uncertainty on the half-life of the parent nuclide in the case of an atom ratio measurement, (3) or the relative uncertainty on the half-life of the daughter nuclide in the case of an activity ratio measurement, (4) while atom ratio measurements are less sensitive to the daughter half-life (5) and activity ratio measurements less sensitive to the parent half-life. Activity ratio measurements enhance the signal of the shortlived nuclide, and may be a good alternative to atom ratio measurements in cases where the atom concentration of the short-lived nuclide is particularly low. An overview of propagation factors for important parent–daughter pairs is presented in Table 3. The relative uncertainty of the half-lives of the parent (via R) or the daughter nuclide (via RA) is of equal importance as the relative uncertainty of the measured ratio (R or RA). In order to achieve a time resolution of 50 days (k ¼1) over a period of 1–50 years, the half-life uncertainties need to be lower than 0.25%, which has not yet been achieved for several important nuclides. Recommended decay data differ between the radionuclide metrology and geochronology communities. More harmonisation is needed in uncertainty evaluation, traceability and reporting style. In particular, one has to be aware that half-life uncertainties are often underestimated, which may implies hidden errors, overly
optimistic confidence levels in age determinations and potentially erroneous conclusions in forensic investigations.
Acknowledgements The authors would like to thank Ken Inn and Jackie Mann of NIST as well as Amy Gaffney and Ross Williams of Lawrence Livermore National Laboratory and Klaus Mayer from EC-JRC-ITU for useful discussion of chronometry of actinides for forensics purposes. They acknowledge the support given by John Keightley at NPL and Evelien Pommé on mathematical issues and appreciate the helpful comments received from a reviewer. References Cheng, H., Edwards, R.L., Hoff, J., Gallup, C.D., Richards, D.A., Asmerom, Y., 2000. The half-lives of 234U and 230Th. Chem. Geol. 169, 17–33. Cheng, et al., 2013. Improvements in 230Th dating, 230Th and 234U half-life values, and U–Th isotopic measurements by multi-collector inductively coupled plasma mass spectrometry. Earth Planet. Sci. Lett. 371-372, 82–91. Croft, S., Burr, T.L., Favalli, A., 2013. Estimating the half-life of 241Pu and its uncertainty. Radiat. Meas. 59 (2013), 94–102. DDEP, 2004-2013. Table of radionuclides. Monographie BIPM-5 BIPM, Sèvres, website: ohttp://www.nucleide.org/DDEP_WG/DDEPdata.htm 4. ENSDF, 2013. Evaluated Nuclear Structure File, NNDC. Brookhaven National Laboratory, o http:/www.nndc.bnl.gov 4. Eppich, G., Williams, R.W., Gaffney, A.M., Schorzman, K.C., 2013. 235U–231Pa age dating of uranium materials for nuclear forensic investigations. J. Anal. At. Spectrom. 28, 666–674. Jaffey, A.H., Flynn, K.F., Glendenin, L.E., Bentley, W.C., Essling, A.M., 1971. Precision measurement of half-lives and specific activities of 235U and 238U. Phys. Rev. C: Nucl. Phys. 4, 1889–1906. Hamilton, E., 1965. Applied Geochronology. Academic Press, ISBN: 0-123-21450-5, pp. 8–9 Magill, J., Galy, J., 2005. Radioactivity Radionuclides Radiation. Springer, ISBN: 3-540-21116-0 Mayer, K., Wallenius, M., Fanhängel, T., 2007. Nuclear forensic science—from cradle to maturity. J. Alloys Compd. 444–445, 50–56. Mayer, K., Wallenius, M., Varga, Z., 2013. Nuclear forensic science: correlating measurable material parameters to the history of nuclear material. Chem. Rev. 113, 884–900. Pommé, S., 2007. Problems with the uncertainty budget of half-life measurements. Applied modeling and computations in nuclear science. T.M. Semkow, S. Pommé, S.M. Jerome, D.J. Strom, (Eds.), In: ACS Symposium Series 945. American Chemical Society, Washington, DC, 2007. 0-8412-3982-7, pp. 282–292. Pommé, S., Camps, J., Van Ammel, R., Paepen, J., 2008. Protocol for uncertainty assessment of half-lives. J. Radioanal. Nucl. Chem. 276, 335–339. Schön, R., Winkler, G., Kutschera, W., 2004. A critical review of experimental data for the half-lives of the uranium isotopes 238U and 235U. Appl. Radiat. Isot 60, 263–273. Schwantes, J.M., Douglas, M., Bonde, S.E., Briggs, J.D., Farmer, O.T., Greenwood, L.R., Lepel, E.A., Orton, C.R., Wacker, J.F., Luksic, A.T., 2009. Nuclear archeology in a bottle: evidence of pre-Trinity U.S. weapons activities from a waste burial site. Anal. Chem. 81, 1297–1306. Steiger, R.H., Jäger, E., 1977. Subcommission on geochronology: convention on the use of decay constants in geo- and cosmochronology. Earth Planet. Sci. Lett. 36, 359–362. United States Court of Appeals, ‘Daubert vs. Merrill-Dow Pharmaceuticals’, Ninth Circuit, (1995), 513–516. Williams, R.W., Gaffney, A.M., 2011. 230Th–234U model ages of some uranium standard reference materials. Proc. Radiochim. Acta 1, 31–35.
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Uncertainty propagation in nuclear forensics S. Pommé a,n, S.M. Jerome b, C. Venchiarutti a a b
European Commission, Joint Research Centre, Institute for Reference Materials and Measurements, Retieseweg 111, B-2440 Geel, Belgium National Physical Laboratory, Teddington, Middlesex TW11 OLW, UK
H I G H L I G H T S
Uncertainty propagation formulae for age dating with nuclear chronometers. Applied to parent–daughter pairs used in nuclear forensics. Investigated need for better half-life data.
art ic l e i nf o
a b s t r a c t
Article history: Received 29 November 2013 Received in revised form 31 January 2014 Accepted 6 February 2014 Available online 14 February 2014
Uncertainty propagation formulae are presented for age dating in support of nuclear forensics. The age of radioactive material in this context refers to the time elapsed since a particular radionuclide was chemically separated from its decay product(s). The decay of the parent radionuclide and ingrowth of the daughter nuclide are governed by statistical decay laws. Mathematical equations allow calculation of the age of specific nuclear material through the atom ratio between parent and daughter nuclides, or through the activity ratio provided that the daughter nuclide is also unstable. The derivation of the uncertainty formulae of the age may present some difficulty to the user community and so the exact solutions, some approximations, a graphical representation and their interpretation are presented in this work. Typical nuclides of interest are actinides in the context of non-proliferation commitments. The uncertainty analysis is applied to a set of important parent–daughter pairs and the need for more precise half-life data is examined. & 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).
Keywords: Nuclear forensics Uncertainty Half-life Decay data Accuracy Non-proliferation
1. Introduction Age dating of radioactive material by means of radiometric or mass spectrometry measurements is a long-established technique in geological and archaeological sciences (Hamilton, 1965; Magill and Galy, 2005). It uses statistical decay laws to provide the link between the activity or atom concentration of radionuclides and their daughter nuclide(s) and the time elapsed since certain initial conditions. Nuclear forensics is a relatively young scientific discipline in which age dating is performed using the same principles. Here, the age refers to the time elapsed since a radionuclide of interest, usually an actinide isotope, was chemically separated from its decay products (Mayer et al., 2007; 2013). Such information taken with other evidence is crucial for identification of the sampled material (see e.g. Schwantes et al. (2009)). The precision by which the age of the material can be determined not only depends on the precision of the analytical measurement,
n
Corresponding author. Tel.: þ 32 14 571 289; fax: þ 32 14 571 864. E-mail address: [email protected] (S. Pommé).
but is also limited by the state of knowledge of the half-lives involved. In this paper, exact mathematical formulae are presented of the uncertainty propagation factors involved with age dating in nuclear forensics, based on the measurement of either atom ratios or activity ratios. From the exact equations, approximate formulae are derived which are applicable under specified conditions. Graphs of the uncertainty propagation as a function of time for two hypothetical cases are presented and interpreted. The equations are used specifically to examine the constraints imposed on the attainable precision of the age due the current uncertainties on the nuclear half-lives of the parent–daughter pairs involved.
2. Age dating by atom ratio measurements 2.1. Exact formulae Age dating is applied to a material containing a radionuclide and its decay products, assuming that no physicochemical processes other than radioactive decay have altered their relative
http://dx.doi.org/10.1016/j.apradiso.2014.02.005 0969-8043 & 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).
S. Pommé et al. / Applied Radiation and Isotopes 89 (2014) 58–64
concentrations over time. The number of atoms of the parent and daughter nuclides, P(t) and D(t) respectively, obey statistical decay rules as a function of time: PðtÞ ¼ Pð0Þe λP t DðtÞ ¼ Dð0Þe λD t þ Pð0Þ
ð1Þ λP ðe λP t e λD t Þ λD λP
ð2Þ
in which the decay constants λP and λD are inversely proportional to the parent and daughter half-lives, TP ¼ln(2)/λP and TD ¼ln(2)/λD, respectively. The ratio of Eq. (2) and Eq. (1) gives the equation for the atom ratio of daughter and parent atoms at any time t40: RðtÞ ¼
DðtÞ λP ¼ Rð0Þe ðλD λP Þt þ ð1 e ðλD λP Þt Þ PðtÞ λD λP
ð3Þ
It is assumed that at time t¼0, the parent nuclei were separated completely from the daughter nuclei such that R(0)¼0, so that D(t) can be fully ascribed to ingrowth after t¼0. Determining the ‘age’ of the material consists of calculating the most likely amount of time elapsed since separation, using measured values of the atom ratio, ^ RðtÞ, and literature values of the decay constants, λ^ P and λ^ D : ! 1 λ^ D λ^ P ^ Age ¼ ln 1 RðtÞ ð4Þ λ^ P λ^ D λ^ P Linear propagation of uncertainty on the atom ratio R(t) and the decay constants λP and λD results to: 2 sðtÞ 2 λD T λP sðλP Þ 2 ¼ t λP λP λD t λD 2 λD T sðλD Þ 2 1 þ λD λP λD t 2 2 T sðRÞ þ ð5Þ t R in which the variable T is defined as T eðλD λP Þt 1 ¼ t ðλD λP Þt
ð6Þ
and the relative uncertainties on the decay constants are equal to (minus) the relative uncertainties on the half-lives, i.e. sðλÞ=λ ¼ sðT 1=2 Þ=T 1=2 : The uncertainty on R arises mainly from the uncertainty of the analytical measurement result, which may be expected to be smallest around R¼1 and to increase in approximate proportion to |log(R)| when parent and daughter concentrations differ by orders of magnitude. Another uncertainty component that should be taken into account is the possibility of an incomplete chemical separation at time t¼ 0 (Williams and Gaffney, 2011; Eppich et al., 2013). This would increase R(t) by a relative amount Rð0Þ=RðtÞe ðλD λP Þt a0 (Eq. (3)), which also propagates by a factor T/t to the relative uncertainty of the age estimate (Eq. (5)).
the daughter nuclide, R(TD) EλP/2λD. As a result, the expectation value of the daughter concentration remains smaller than that of the parent concentration. In the case of a long-lived daughter, where λD⪡λP, the atom ratio reaches unity after about one half-life of the parent, R(TP)E1, and then increases exponentially by RðtÞ eλP t for λPt 41. 2.2.2. Uncertainty propagation The uncertainty propagation formulae in Eq. (5) can be stated as a series expansion, resulting in approximate formulae valid for |λD λP|to1: 2 sðtÞ 2 λD t λP ðλD λP Þt 2 sðλP Þ 2 1 t 2 λP 6 2 2 2 λD t λP ðλD λP Þt sðλD Þ þ þ 2 λD 6 !2 2 2 ðλD λP Þt ðλD λP Þ t sðRÞ 2 þ þ 1þ ð7Þ 2 R 6
In Table 1, approximate equations are summarised for the propagation factors under boundary conditions. For material with a relatively young age compared to the half-lives involved, λPt⪡1 and λDt⪡1, the propagation factors for λP and R are unity, while the factor for λD is insignificantly small. Consequently, age determinations based on atom ratios are more sensitive to the parent half-life then to the daughter half-life. This is compatible with the fact that R(t) EλPt for small values of t, i.e. linear with λP and independent of λD. Only for old material, |λD λP|t b 1, does the uncertainty propagation of the daughter half-life become important, but under these conditions the over-all accuracy of the method is relatively poor, as the propagation factors increase almost exponentially with λDt. 2.3. Graphical representation The propagation factors are presented for two hypothetical cases: a long-lived parent nuclide (λP⪡λD) in Fig. 1 and a long-lived daughter nuclide (λD⪡λP) in Fig. 2. 2.3.1. Long-lived parent In the top graph of Fig. 1, there is no visible difference between the dominant propagation factors of λP and R. They remain close to unity as long as λDto1. For larger values of t, the propagation factors increase exponentially with λDt. The age dating method Table 1 Exact and approximate equations for uncertainty propagation factors for age dating via atom ratio measurements in specific conditions. The variable T has been defined in Eq. (6). Exact equations are also presented as a function of the half-lives TP and TD instead of the decay factors λP and λD. Condition
2.2. Approximate equations Always valid
2.2.1. Atom ratio Serial expansion of Eq. (3) shows that R(t) EλPt if the age of the material is low compared to the half-lives of the nuclides involved (|λD λP|t o1). This means that the initial ingrowth of the daughter happens at the rate of the decay of the parent and is, to a first order approximation, independent of the half-life of the daughter. Further changes to R(t) with time depend on which nuclide has the longest half-life. In the case of a long-lived parent, where λP⪡λD, the atom ratio tends towards the secular equilibrium value R(t)E λP/λD for λDt 41 and reaches about half that value after one half-life of
59
Always valid
Age from atom ratio, using Eq. (3) Propagation factor ∂t λP ∂λP t λD T λP λP λD t λD TP T D T P Tt TT DP
λP { λD λD t{1
Long-lived parent 1
λD t c 1 λP c λD λP t{1
eλD t λD t Long-lived daughter 1
λP t c 1
1
∂t λD ∂λD t λD T λP λD t 1 T P T T D T P t 1
∂t R
λD t 2 eλD t λD t
1
λD t 2 λD λP
∂R t
T t
T t
eλD t λD t 1 1 λP t
60
S. Pommé et al. / Applied Radiation and Isotopes 89 (2014) 58–64
Fig. 1. Hypothetical case of a long-lived parent nuclide with 100 times longer halflife than its daughter nuclide. Top: Propagation factors that relate relative uncertainty of age dating by means of atom ratio measurements with relative uncertainties on the analytical result and on the decay constants of the parent and daughter nuclide (Eq. (5)). Middle: Uncertainty propagation factors for activity ratio measurements (Eq. (12)). Bottom: Order of magnitude of the atom and activity ratio (full line) and indicator of uncertainty on the analytical result (cf. Eq. (15)) multiplied with the propagation factor T/t (dotted line). The latter can be used to roughly identify a time region in which age dating can be performed accurately.
breaks down because the atom ratio between parent and daughter concentration tends towards the secular equilibrium value of λP/(λD λP). Under these conditions, where λDt⪢1, the method can only establish a minimum age value, i.e. Age⪢1/λD. 2.3.2. Long-lived daughter The top graph of Fig. 2 refers to the case of a long-lived daughter nuclide (λD⪡λP). The propagation factor of λP is about 1 and that of λD is significantly lower, with values varying between λDt/2 (for λPto1) and λD/λP (for λPt41). The propagation factor for R is close to 1 for λPto1 and decreases quasi-linearly towards 1/λPt for λPt41. The method seemingly gains accuracy with time; however, one also has to take the uncertainty s(R) of the atom ratio into account. After about one half-life of the parent nuclide, the concentration of the daughter nuclide becomes greater than that of the parent nuclide and one expects the uncertainty of the analytical measurement, s(R)/R, to increase as this ratio grows exponentially with λPt over many orders of magnitude. The method will break down at a point where the ratio R becomes too large to be measurable. The precision of the age may at all times be limited by the level of uncertainty on the half-life of the parent.
Fig. 2. Hypothetical case of a short-lived parent nuclide with 100 times shorter half-life than its daughter nuclide. Top: Propagation factors that relate relative uncertainty of age dating by means of atom ratio measurements with relative uncertainties on the analytical result and on the decay constants of the parent and daughter nuclide (Eq. (5)). Middle: Uncertainty propagation factors for activity ratio measurements (Eq. (12)). Bottom: Order of magnitude of the atom and activity ratio (full line) and indicator of uncertainty on the analytical result (cf. Eq. (15)) multiplied with the propagation factor T/t (dotted line). The latter can be used to roughly identify a time region in which age dating can be performed accurately.
constants, i.e. AP(t)¼P(t)λP and AD(t)¼D(t)λD. The time dependence follows from the statistical decay rules: AP ðtÞ ¼ AP ð0Þe λP t
ð8Þ
λD AD ðtÞ ¼ AD ð0Þe λD t þ AP ð0Þ ðe λP t e λD t Þ λD λP
ð9Þ
and the activity ratio of a parent–daughter pair at time t is RA ðtÞ ¼
AD ðtÞ λD ¼ RA ð0Þe ðλD λP Þt þ ð1 e ðλD λP Þt Þ AP ðtÞ λD λP
ð10Þ
Again one assumes full separation between parent and daughter nuclides at time t ¼0, such that RA(0) ¼0. The ‘age’ of the material can be calculated from a measured activity ratio value, R^ A ðtÞ, and best estimates of the decay constants, λ^ P and λ^ D : ! 1 λ^ D λ^ P Age ¼ ln 1 R^ A ðtÞ ð11Þ λ^ P λ^ D λ^ D
3. Age dating by activity ratio measurements 3.1. Exact equations The activities of the parent and daughter nuclides in a material are proportional to their respective number of atoms and decay
The uncertainty propagation of the concentration ratio RA(t) and the decay constants λP and λD to the age is given by 2 sðtÞ 2 λP T sðλP Þ 2 1 ¼ t λP λP λD t
S. Pommé et al. / Applied Radiation and Isotopes 89 (2014) 58–64
λP T λD sðλD Þ λD λP λD t λP T sðRA Þ þ t RA
3.3. Graphical representation
þ
ð12Þ
in which the variable T is the same as in Eq. (6). The uncertainty on RA mainly arises from the uncertainty of the activity ratio measurement result, which may be expected to be at its smallest around RA ¼ 1 and to grow roughly proportional with |log(RA)| when parent and daughter activities differ by orders of magnitude. 3.2. Approximate equations 3.2.1. Activity ratio For small values of t, where |λD λP|to1, the activity ratio in Eq. (10) reduces to RA(t) EλDt. This is a consequence of the fact that the atom ratio grows linearly with time (R(t)EλPt) and the specific activity ratio between same amounts of daughter and parent nuclide is λD/λP. The rate at which the daughter activity initially grows with time is, to a first order approximation, independent of the half-life of the parent. Old material, in the case of a long-lived parent (λP⪡λD), tends towards secular equilibrium in which parent and daughter activities are nearly the same, RA E1, whereas in the case of a long-lived daughter (λD⪡λP), the activity ratio increases exponentially, as RA ðtÞ λD =λP eλP t for λPt41. 3.2.2. Uncertainty propagation The uncertainty propagation formulae in Eq. (12) for age dating by activity ratio can also be stated as a series expansion: 2 sðtÞ 2 λP t λP ðλD λP Þt 2 sðλP Þ 2 t 2 λP 6 2 λP t λP ðλD λP Þt 2 sðλD Þ 2 þ þ 1 þ 2 λD 6 !2 2 2 ðλD λP Þt ðλD λP Þ t sðRA Þ 2 þ þ 1þ ð13Þ 2 RA 6 In Table 2, approximate equations are summarised for the propagation factors under boundary conditions. For material with a young age compared to the half-lives involved, λPt⪡1 and λDt⪡1, the propagation factors for λD and RA are unity, while the factor for λP is insignificantly small. Consequently, age determinations based on activity ratios are more sensitive to the daughter half-life then to the parent half-life, which is the opposite of the effect noticed for atom ratios (Eq. (7)). Table 2 Exact and approximate equations for uncertainty propagation factors for age dating via activity ratio measurements in specific conditions. The variable T has been defined in Eq. (6). Exact equations are also presented as a function of the half-lives TP and TD. Condition
Always valid Always valid λP {λD λD t{1 λD t c 1 λP cλD λP t{1 λP t c 1
Age from activity ratio, using Eq. (11) Propagation factor ∂t λP ∂λP t λP T λP λD t 1 T D T T D T P t 1 Long-lived parent λP t 2 λP eλD t λD λD t Long-lived daughter
61
∂t λD ∂λD t λP T λD λP λD t λP TD T TP T D T P t T D
∂t RA ∂RA t T
1
1 λD t
t
T t
λP e λD λD t
eλD t λD t
λP t 2
1
1
1
1 λP t
1 λP t
To interpret the impact of various parameters on the over-all uncertainty of age dating via the activity ratio, one can refer again to Figs. 1 and 2 representing two hypothetical cases: a long-lived parent nuclide, when λP⪡λD, in Fig. 1 and a long-lived daughter nuclide, when λD⪡λP, in Fig. 2. 3.3.1. Long-lived parent In the middle graph of Fig. 1, the dominant propagation factors of λD and RA remain close to unity so long as λDto 1. The propagation factor of λP is much smaller, but increases exponentially for λDt⪢1 and eventually its uncertainty propagation becomes as important as that of λD, but inferior by a factor of λP/ λD to the propagation factor of RA. For large values of t, the method can only establish a minimum age value, i.e. Age⪢1/λD. 3.3.2. Long-lived daughter The middle graph of Fig. 2 refers to the case of a long-lived daughter nuclide, (λD⪡λP). The propagation factors of RA and λD are nearly equal: close to unity for λPto 1 and linearly decreasing with 1/λPt for λPt 41. The propagation factor of λP is small for λPto 1 and tends to unity for λPt 41.
4. Atom versus activity ratio 4.1. Indicator of accuracy It is interesting to note that the activity ratio can differ significantly from the atom ratio if parent and daughter nuclides have distinctly different half-lives. The choice of the most suitable measurement technique for a given parent–daughter pair may be evaluated by comparison through the relationship log ðRA Þ ¼ log λD þ log ðRÞ ð14Þ λP The activity ratio measurements inflate the signal of the shortlived nuclide, which is an analytical advantage in the case of a long-lived parent (log(λD/λP) 40), to compensate for the relatively low concentration of a short-lived daughter (log(R) o0). It offers also potential with aged material with a short-lived parent (log(λD/ λP) 40), to amplify the signal of its remaining small traces (log(R) 40). The bottom graphs in Figs. 1 and 2 show the logarithmic value of R and RA, as well as a new quantity:
1 T T log 2þRþ j log ðRÞj þ log ð15Þ R t t which is assumed to be roughly indicative of the relative uncertainty s(R)/R and its uncertainty propagation factor T/t towards the age. It was derived from an assumption that the signals from D and P are Poissonian, such that the relative uncertainty on R would be inversely proportional with the count rates, i.e. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi log ðsðRÞ=RÞ p log ð 1=P þ 1=DÞ p log ðð1 þ RÞ þ ð1 þ 1=RÞÞ, since P/(P þD) ¼1/(1 þ R) and D/(Pþ D)¼R/(1þR). A similar quantity for the activity ratio RA was plotted in Figs. 1 and 2. The lower values of these quantities roughly indicate the time range in which the age dating methods should be relatively accurate. 4.2. Long-lived parent In Fig. 1, the activity ratio is closer to unity than the atom ratio over the whole time range, which in principle is advantageous. Whether activity measurements can lead to a more accurate age
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determination will depend on the difference in attainable precision by radiometric methods and mass spectrometry, as well as the difference in accuracy of the half-lives of parent and daughter nuclide involved. The long half-life of the parent implies a low specific activity, which may be problematic when measuring the activity of small amounts of material. In the bottom graph of Fig. 1, the time region in which both age dating methods are likely to be more accurate roughly corresponds to ages varying within 1/10 and 3 times the half-life of the daughter nuclide TD. 4.3. Long-lived daughter In the bottom graph of Fig. 2, the most favourable time regions of both methods differ significantly. Atom ratio measurements should be informative over a wide time region, for example between 10 3 and 10 times the half-life of the parent nuclide TP. Activity ratios are less favourable chronometers for this type of decay, unless for old material, where the age is between 1/3 and 12 times TP.
half-lives are often unrealistically low (Pommé, 2007; Pommé et al., 2008). As an alternative to atom ratio measurements, activity measurements offer – at least in principle – a lower minimum uncertainty for certain chronometers, like 232U–228Th, 238 U–234Th, 237Np–233U, 242Pu–238U and 243Am–239Pu. Whether this can also be realised in practice heavily depends on the achievable precision on the activity ratio, as the radiometric method may have a lower dynamic range than mass spectrometry. The reader is referred to Mayer et al. (2013) for a discussion of measurement techniques for U and Pu chronometers. 5.3. Other decay types 5.3.1. Stable daughter In Table 3, one finds decay types that do not strictly match the parent–daughter decay model investigated in this work. Some of the daughter nuclides are stable (λD ¼0) and the equation for the age dating via atom ratio reduces to: Age ¼
1 ^ lnð1 þ RðtÞÞ λ^ P
ð16Þ
5. Specific radionuclide chronometers 5.1. Nuclear chronometers Nuclear forensics refers to the scientific analysis of radioactive material in the context of legal procedures (Mayer et al., 2013), and is no different in concept than any other forensic investigation. One may think of material used in the construction of improvised nuclear devices (IND) or radiation dispersion devices (RDD) that may be detonated in populated areas. Age dating is part of characterisation of nuclear material, which may play a role in establishing the origin of intercepted material, identifying perpetrators and their network, as well as providing evidence to bring them to justice. Given the severity of such activities and the possibly long-reaching consequences, it is imperative that all parts of the forensic analysis comply with the accepted norms for the presentation of scientific evidence in court (United States Appeal Court, 1995). The chronometers of most interest are actinides arising from their potential use in INDs, which includes isotopes of thorium (Z¼ 90), uranium (Z¼92), neptunium (Z¼93), plutonium (Z¼94) and americium (Z ¼95), whereas candidate materials for RDDs may include radionuclides such as 60Co, 89Sr, 90Sr and 137Cs. Table 3 shows a non-exhaustive list of nuclear chronometers, the evaluated half-lives of the parent–daughter pairs (DDEP, 2004– 2013), the uncertainty propagation factors (cf. equations in Tables 1 and 2) for material of age 1 year and 50 years, and a minimum uncertainty value based on the propagation of the halflife uncertainties only. 5.2. Objective If one sets as an objective to keep the uncertainty on age dating below 50 days over a range of 1–50 years, which corresponds to a relative uncertainty of 0.27% (k ¼1), then the relative uncertainties on the ratio measurements and the half-lives should be below 0.25%. In Table 3, the dominant uncertainty component is indicated in bold when assuming hypothetically that the uncertainty on the ratio measurement is 0.25%. Any nuclide of which the propagation factor is in bold therefore would need a more accurate half-life value to achieve the objective of s(Age) o50 days. This includes a majority of the nuclides in the list, of which 232 U, 229Th and 242mAm are obvious examples. Moreover, the list may in reality be longer, as the uncertainty estimates on measured
Obviously the relative uncertainty on the half-life of the parent is propagated linearly to the age, hence the propagation factor for λP is unity for 60Co, 89,90Sr and 137Cs (cf. Table 3). 5.3.2. Parent–daughter–granddaughter In some other cases, the decay type is more complex and involves three subsequent decays. Examples of parent–daughter– granddaughter decays are 90Sr–90Y–90Zr, 237Np–233Pa–233U, 242m Am–242gAm–242Pu and 243Am–239Np–239Pu. In these specific cases, the half-life of the daughter – 2.7 d (90Y), 27 d (233Pa), 16 h (242gAm) and 2.4 d (239Np) – is very small compared to that of the parent and granddaughter. As a result, they hardly delay the decay from parent to granddaughter and the parent–granddaughter pair can be regarded as a simple parent–daughter decay when λDt⪢1. This can be verified from the statistical equation for the atom concentration of the granddaughter nuclide: λD GðtÞ ¼ Gð0Þe λG t þ Dð0Þ ðe λD t e λG t Þ λG λD λP λD λP λD λP λD e λP t þ e λD t þ e λG t þ Pð0Þ λD λP λG λP λP λD λG λD λP λG λD λG ðGð0Þ þ Dð0ÞÞe λG t þPð0Þ
λP λD t λP e þ Pð0Þ ðe λP t e λG t Þ λD λG λP
ð17Þ
The concentration D(t) of the short-lived daughter is always small compared to P(t); their atom ratio is proportional to λP/λD. The concentration of the granddaughter, G(t), is hardly influenced by the daughter, as the term with λD is proportional to λP/λD and contains an exponential factor which is small for λDt41. Thus, for ages larger than the (short) half-life of the daughter, Eq. (17) reduces to Eq. (2) in which the granddaughter takes the place of the daughter. The uncertainty propagation of λD to the determined age is negligible. 5.4. Recommended data 5.4.1. Data evaluation Since the radionuclide community has to consider, use and thus recommend the half-lives of many different radionuclides, considerable effort has gone into evaluating nuclear decay data for an increasing number of radionuclides under the Decay Data Evaluation Project (DDEP, 2004-2013). Where a radionuclide is not yet included in the DDEP database, the usual recommendation is to
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63
Table 3 Parent–daughter pairs usable as nuclear chronometers, their half-lives and propagation factors for the relative uncertainty on age dating in a range of 1–50 years. Most halflife data are taken from DDEP (2004-2013) and some from ENSDF (2013). Propagation factors in bold indicate a dominant uncertainty component in the hypothetical case that s(R)/R ¼ s(RA)/RA ¼0.25%. The minimum uncertainties on the age dating, umin(days), after 1 year and after 50 years are calculated taking into account only the uncertainties on the half-lives (adopting s(R)/R¼ 0%.). Parent A
Z
60
Co Sr 90 Sr 137 Cs 230 Th 232 Th 232 U 233 U 234 U 235 U 236 U 238 U 237 Np 236 Pu 238 Pu 239 Pu 240 Pu 241 Pu 242 Pu 241 Am 242m Am 243 Am 89
n
Daughter T 1=2 ðaÞ
sðT 1=2 Þ
5.2711 (8) 0.13846 (8) 28.80 (7) 30.05 (8) 7.538 (30) 104n 1.402 (6) 1010 70.6 (1.1) 1.592 (2) 105n 2.455 (6) 105 7.04 (1) 108 2.343 (6) 107n 4.468 (5) 109 2.144 (7) 106 2.858 (8)n 87.74 (3) 24100 (11) 6561 (7) 14.33 (4) 3.73 (3) 105 432.6 (6) 143 (2) 7367 (23)
0.015 0.06 0.24 0.27 0.4n 0.43 1.6 0.13n 0.24 0.14 0.26n 0.11 0.33 0.28n 0.034 0.05 0.11 0.28 0.8 0.14 1.4 0.31
T 1=2
(%)
A
Z
60
Ni Y 90 Zr 137 Ba 226 Ra 228 Ra 228 Th 229 Th 230 Th 231 Pa 232 Th 234 Th 233 U 232 Th 234 U 235 U 236 U 241 Am 238 U 237 Np 242 Pu 239 Pu 89
Age via atom ratio R T 1=2 ðaÞ
sðT 1=2 Þ
1 1 1 1 1600 (7) 5.75 (4) 1.9126 (9) 7.34 (16) 103n 7.538 (30) 104n 3.267 (26) 104 1.402 (6) 1010 6.600 (8) 10 2 1.592 (2) 105n 1.402 (6) 1010 2.455 (6) 105 7.04 (1) 108 2.343 (6) 107 423.6 (6) 4.468 (5) 109 2.144 (7) 106 3.73 (3) 105 24100 (11)
– – – – 0.44 0.70 0.05 2.2n 0.4n 0.8 0.43 0.12 0.13n 0.43 0.24 0.14 0.26 0.14 0.11 0.33 0.8 0.8
T 1=2
(%)
Age via activity ratio RA
Combined
λP
λD
umin(d)
λP
λD
umin(d)
R, RA
1 1 1 1 1 1–70 1.2–1 1 1 1 1 3500–1 1 1 1 1 1 1 1 1 1 1
– – – – 0–0.01 0.06–70 0.2–1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.1–3 0.2–11 0.9–44 1–49 1.5–74 2–1E4 7–1 0.5–23 0.9–45 0.5–26 0.9–47 2E3–1 1–60 1–51 0.1–6 0.2–8 0.4–20 1–52 3–147 0.5–25 5–255 1–57
– – – – 0 0 0–1 0 0 0 0 0 0 0.1–0.9 0–0.2 0 0 0–0.6 0–150 0 0–0.1 0
– – – – 1 1–70 1–1 1 1 1 1 1 1 0.9–0.08 1–0.8 1 1 1–0.4 1 1 1–0.9 1
– – – – 2–80 3–127 0.2–1 8–398 1.5–73 3–145 2–78 0.5–1 0.5–23 1–47 0.9–37 0.5–26 0.9–47 0.5–34 0.4–20 1–57 3–133 0.2–8
0.9–0.15 0.2–0.004 1–0.6 1–0.6 1 1–70 1.2–1 1 1 1 1 3500–1 1 0.9–0.08 1–0.8 1 1 1–0.4 1 1 1–0.9 1
Data from ENSDF (2013).
seek data from the Evaluated Nuclear Structure File database, also published on line (ENSDF, 2013). The aim of evaluation of data is to provide the best estimate of the value of the half-live and a realistic estimate of its uncertainty.
Table 4 Comparison of half-life values and standard uncertainties (k ¼ 1) in geo- and cosmochronology and in radionuclide metrology. In the case of 230Th and 234U, recent data were included for a discussion of uncertainties (see Section 5.4.4). Nuclide
5.4.2. User community There is, however, some difference between the half-life data sets used by the radionuclide metrology community and the geochemistry community. By comparison, the geochemistry community have a smaller list of radionuclides to consider and has an interest in using a fixed, common data set for reasons of traceability and comparability of age dating results among various studies, with constraints on consistency and ‘…all selected values are open to and should be the subjects of continuing critical scrutinizing and laboratory investigation…’ (Steiger and Jäger, 1977). A partial comparison between the two sets of recommended data is given in Table 4, in which we included data from Steiger and Jäger (1977) and recently measured half-life values for 234U and 230Th (Cheng et al., 2013). The half-life values are fairly consistent, as they often rely on the same key measurements. This is for instance the case with 235,238U, which predominantly relies on the work of Jaffey et al. (1971). However, there is an essential difference in the treatment of uncertainties. Data evaluators have a more critical approach to the low uncertainties claimed in the past, resulting in a much higher uncertainty estimate (Schön et al., 2004; DDEP, 2004-2013). These evaluated data have not been applied in the geochronology community and therefore the low uncertainty estimates of Jaffey et al. (1971) have been propagated to daughter nuclides 234U and 230Th. 5.4.3. Problems with uncertainty Considering that nuclear forensics will be subject to scrutiny in a court of law, it is essential that it relies on the best available data set and in particular that it applies realistic uncertainties to avoid overly optimistic claims. One needs to be aware of problems
Geo- and cosmochronology T 1=2 ðaÞ
40
K Th 232 Th 234 U 235 U 238 U 230
1.2505 109a 7.5584 (55)103c 1.40100 1010a 2.45620 (13) 105c 7.03810 108a 4.46831 109a
sðT 1=2 Þ T 1=2
0.07 0.05
(%)
Radionuclide metrology T 1=2 ðaÞ
sðT 1=2 Þ
1.2504 (30) 109b 7.538 (30) 104d 1.402 (6) 1010a 2.455 (6) 105b 7.04 (1) 108b 4.468 (5) 109b
0.24 0.40 0.43 0.24 0.14 0.11
T 1=2
(%)
a
Steiger and Jäger (1977)—no stated uncertainty. DDEP (2004-2013). Cheng et al. (2013). d ENSDF (2013). b c
related to the assignment of uncertainties to half-lives (Pommé, 2007; Croft et al., 2013) and it is essential that future publications of new decay data would include a realistic uncertainty calculation and apply a complete reporting style to improve consistency of data sets and achieve traceability on how they were produced (Pommé et al., 2008). One of the minimum requirements is to provide a table with the uncertainty budget, i.e. a list of all possible uncertainty components, their estimated value and propagation to the half-life uncertainty. The accompanying text should clarify how these uncertainties were calculated or estimated.
5.4.4. Example Cheng et al. (2000; 2013) determined the half-lives of 234U and 230 Th twice, by measuring atom ratios in old materials and assuming that both daughter nuclides are in secular equilibrium with their long-lived parent 238U. They calculated relative uncertainty values of 0.053% (k¼ 1) and 0.073% (k ¼1) for T1/2(234U) and
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T1/2(230Th), respectively, which are dominated by the 0.054% relative uncertainty estimate on T1/2(238U) by Jaffey et al. (1971). The uncertainty estimates on the 234U/238U and 230Th/238U atom ratios are lower: 0.017% (k¼ 1) and 0.053% (k ¼1), respectively. However, their atom ratio results from 2013 differ by 0.14% and 0.14% with the ratios that they published in 2000, with uncertainties of 0.086% (k¼ 1) and 0.050% (k ¼1). The problem arises that the stated uncertainties of the 234U and 230Th may be underestimated, since: (1) the uncertainty on T1/2(238U) may be unrealistic (Schön et al., 2004; DDEP, 20042013), (2) the uncertainty propagation should have yielded relative uncertainties of 0.056% and 0.076% instead of 0.053% and 0.073%, (3) the uncertainties on the atom ratio values are barely sufficient to cover differences of 0.14% between data obtained in 2000 and 2013 by the same authors, (4) there may be additional systematic errors involved that do not show up by repetition of the same type of measurement. 6. Conclusions Exact and approximate uncertainty propagation factors have been derived mathematically for age dating of nuclear material via atom and activity ratio measurements. They are summarised in Tables 1 and 2 and graphically presented in Figs. 1 and 2. If the age of the material is low compared to the half-life of the nuclides involved, the mathematical equations simplify: (1) the relative uncertainty of the atom or activity ratio measurements propagate by a factor of 1, (2) and so does the relative uncertainty on the half-life of the parent nuclide in the case of an atom ratio measurement, (3) or the relative uncertainty on the half-life of the daughter nuclide in the case of an activity ratio measurement, (4) while atom ratio measurements are less sensitive to the daughter half-life (5) and activity ratio measurements less sensitive to the parent half-life. Activity ratio measurements enhance the signal of the shortlived nuclide, and may be a good alternative to atom ratio measurements in cases where the atom concentration of the short-lived nuclide is particularly low. An overview of propagation factors for important parent–daughter pairs is presented in Table 3. The relative uncertainty of the half-lives of the parent (via R) or the daughter nuclide (via RA) is of equal importance as the relative uncertainty of the measured ratio (R or RA). In order to achieve a time resolution of 50 days (k ¼1) over a period of 1–50 years, the half-life uncertainties need to be lower than 0.25%, which has not yet been achieved for several important nuclides. Recommended decay data differ between the radionuclide metrology and geochronology communities. More harmonisation is needed in uncertainty evaluation, traceability and reporting style. In particular, one has to be aware that half-life uncertainties are often underestimated, which may implies hidden errors, overly
optimistic confidence levels in age determinations and potentially erroneous conclusions in forensic investigations.
Acknowledgements The authors would like to thank Ken Inn and Jackie Mann of NIST as well as Amy Gaffney and Ross Williams of Lawrence Livermore National Laboratory and Klaus Mayer from EC-JRC-ITU for useful discussion of chronometry of actinides for forensics purposes. They acknowledge the support given by John Keightley at NPL and Evelien Pommé on mathematical issues and appreciate the helpful comments received from a reviewer. References Cheng, H., Edwards, R.L., Hoff, J., Gallup, C.D., Richards, D.A., Asmerom, Y., 2000. The half-lives of 234U and 230Th. Chem. Geol. 169, 17–33. Cheng, et al., 2013. Improvements in 230Th dating, 230Th and 234U half-life values, and U–Th isotopic measurements by multi-collector inductively coupled plasma mass spectrometry. Earth Planet. Sci. Lett. 371-372, 82–91. Croft, S., Burr, T.L., Favalli, A., 2013. Estimating the half-life of 241Pu and its uncertainty. Radiat. Meas. 59 (2013), 94–102. DDEP, 2004-2013. Table of radionuclides. Monographie BIPM-5 BIPM, Sèvres, website: ohttp://www.nucleide.org/DDEP_WG/DDEPdata.htm 4. ENSDF, 2013. Evaluated Nuclear Structure File, NNDC. Brookhaven National Laboratory, o http:/www.nndc.bnl.gov 4. Eppich, G., Williams, R.W., Gaffney, A.M., Schorzman, K.C., 2013. 235U–231Pa age dating of uranium materials for nuclear forensic investigations. J. Anal. At. Spectrom. 28, 666–674. Jaffey, A.H., Flynn, K.F., Glendenin, L.E., Bentley, W.C., Essling, A.M., 1971. Precision measurement of half-lives and specific activities of 235U and 238U. Phys. Rev. C: Nucl. Phys. 4, 1889–1906. Hamilton, E., 1965. Applied Geochronology. Academic Press, ISBN: 0-123-21450-5, pp. 8–9 Magill, J., Galy, J., 2005. Radioactivity Radionuclides Radiation. Springer, ISBN: 3-540-21116-0 Mayer, K., Wallenius, M., Fanhängel, T., 2007. Nuclear forensic science—from cradle to maturity. J. Alloys Compd. 444–445, 50–56. Mayer, K., Wallenius, M., Varga, Z., 2013. Nuclear forensic science: correlating measurable material parameters to the history of nuclear material. Chem. Rev. 113, 884–900. Pommé, S., 2007. Problems with the uncertainty budget of half-life measurements. Applied modeling and computations in nuclear science. T.M. Semkow, S. Pommé, S.M. Jerome, D.J. Strom, (Eds.), In: ACS Symposium Series 945. American Chemical Society, Washington, DC, 2007. 0-8412-3982-7, pp. 282–292. Pommé, S., Camps, J., Van Ammel, R., Paepen, J., 2008. Protocol for uncertainty assessment of half-lives. J. Radioanal. Nucl. Chem. 276, 335–339. Schön, R., Winkler, G., Kutschera, W., 2004. A critical review of experimental data for the half-lives of the uranium isotopes 238U and 235U. Appl. Radiat. Isot 60, 263–273. Schwantes, J.M., Douglas, M., Bonde, S.E., Briggs, J.D., Farmer, O.T., Greenwood, L.R., Lepel, E.A., Orton, C.R., Wacker, J.F., Luksic, A.T., 2009. Nuclear archeology in a bottle: evidence of pre-Trinity U.S. weapons activities from a waste burial site. Anal. Chem. 81, 1297–1306. Steiger, R.H., Jäger, E., 1977. Subcommission on geochronology: convention on the use of decay constants in geo- and cosmochronology. Earth Planet. Sci. Lett. 36, 359–362. United States Court of Appeals, ‘Daubert vs. Merrill-Dow Pharmaceuticals’, Ninth Circuit, (1995), 513–516. Williams, R.W., Gaffney, A.M., 2011. 230Th–234U model ages of some uranium standard reference materials. Proc. Radiochim. Acta 1, 31–35.